Steklov Eigenvalues of Submanifolds with Prescribed Boundary in Euclidean Space

Colbois, Bruno; Girouard, Alexandre; Gittins, Katie

doi:10.1007/s12220-018-0063-x

Cited by 26 publications

(39 citation statements)

References 21 publications

(39 reference statements)

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“…The proof of Proposition A.1 can be carried out in the same way as that of [9, Proposition 2.1] (see also [10,Theorem 1.5]). In particular it makes use of Lemma 3.3 here above.…”

Section: Appendix a Isoperimetric Bounds For Compact Hypersurfacesmentioning

confidence: 99%

Conformal upper bounds for the eigenvalues of the p‐Laplacian

Colbois

Provenzano

2021

Journal of London Math Soc

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In this note we present upper bounds for the variational eigenvalues of the p-Laplacian on smooth domains of complete n-dimensional Riemannian manifolds and Neumann boundary conditions, and on compact (boundaryless) Riemannian manifolds. In particular, we provide upper bounds in the conformal class of a given metric for 1 < p n, and upper bounds for all p > 1 when we fix a metric. To do so, we use a metric approach for the construction of suitable test functions for the variational characterization of the eigenvalues. The upper bounds agree with the well-known asymptotic estimate of the eigenvalues due to Friedlander. We also present upper bounds for the variational eigenvalues on hypersurfaces bounding smooth domains in a Riemannian manifold in terms of the isoperimetric ratio.

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“…The proof of Proposition A.1 can be carried out in the same way as that of [9, Proposition 2.1] (see also [10,Theorem 1.5]). In particular it makes use of Lemma 3.3 here above.…”

Section: Appendix a Isoperimetric Bounds For Compact Hypersurfacesmentioning

confidence: 99%

Conformal upper bounds for the eigenvalues of the p‐Laplacian

Colbois

Provenzano

2021

Journal of London Math Soc

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“…The interplay of these eigenvalues with the geometry of M has been an active area of investigation in recent years. See [20] for a survey and [11,12,15,23,4,19,18] for recent relevant results.…”

Section: Introductionmentioning

confidence: 99%

“…They do not directly involve the curvature. See [24,21,13] for early use of similar techniques and [7,8,11] some more recent results in the same spirit. Upper bounds for the eigenvalues λ k of the Laplacian on a closed Riemannian manifold will also be obtained.…”

Section: Introductionmentioning

confidence: 99%

Metric upper bounds for Steklov and Laplace eigenvalues

Colbois,

Girouard

2021

Preprint

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We prove two upper bounds for the Steklov eigenvalues of a compact Riemannian manifold with boundary. The first involves the volume of the manifold and of its boundary, as well as packing and volume growth constants of the boundary and its distortion. The second bound is in terms of the intrinsic and extrinsic diameters of the boundary, as well as its injectivity radius. By applying these bounds to cylinders over closed manifold, we obtain new bounds for eigenvalues of the Laplace operator on closed manifolds, in the spirit of Grigor'yan-Netrusov-Yau and of Berger-Croke. For a family of manifolds that has uniformly bounded volume and boundary of fixed intrinsic geometry, we deduce that a large first nonzero Steklov eigenvalue implies that each boundary component is contained in a ball of small extrinsic radius. This is then contrasted with another concentration of measure phenomenon, akin to Gromov-Milman concentration for closed manifolds.

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“…Discussion and previous results. Quantitative estimates relating individual Steklov eigenvalues to eigenvalues of the tangential Laplacian ∆ Σ have been studied in [28,6,2,3,19,30,25,29]. They are relatively easy to obtain if the manifold M is isometric (or quasi-isometric with some control) to a product near its boundary.…”

Section: Introductionmentioning

confidence: 99%

The Steklov and Laplacian spectra of Riemannian manifolds with boundary

Colbois

Girouard

Hassannezhad

2020

Journal of Functional Analysis

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Given two compact Riemannian manifolds M 1 and M 2 such that their respective boundaries Σ 1 and Σ 2 admit neighbourhoods Ω 1 and Ω 2 which are isometric, we prove the existence of a constant C such that |σ k (M 1 ) − σ k (M 2 )| ≤ C for each k ∈ N. The constant C depends only on the geometry of Ω 1 ∼ = Ω 2 . This follows from a quantitative relationship between the Steklov eigenvalues σ k of a compact Riemannian manifold M and the eigenvalues λ k of the Laplacian on its boundary. Our main result states that the difference |σ k − √ λ k | is bounded above by a constant which depends on the geometry of M only in a neighbourhood of its boundary. The proofs are based on a Pohozaev identity and on comparison geometry for principal curvatures of parallel hypersurfaces. In several situations, the constant C is given explicitly in terms of bounds on the geometry of Ω 1 ∼ = Ω 2 .1991 Mathematics Subject Classification. 35P15 (primary), 58C40, 35P20 (secondary).1 The notation O(k −∞ ) designates a quantity which tends to zero faster than any power of k.

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Steklov Eigenvalues of Submanifolds with Prescribed Boundary in Euclidean Space

Cited by 26 publications

References 21 publications

Conformal upper bounds for the eigenvalues of the p‐Laplacian

Conformal upper bounds for the eigenvalues of the p‐Laplacian

Metric upper bounds for Steklov and Laplace eigenvalues

The Steklov and Laplacian spectra of Riemannian manifolds with boundary

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